Threefold Flops via Matrixfactorization

نویسندگان

  • Carina Curto
  • David Morrison
چکیده

The explicit McKay correspondence, as formulated by GonzalezSprinberg and Verdier, associates to each exceptional divisor in the minimal resolution of a rational double point, a matrix factorization of the equation of the rational double point. We study deformations of these matrix factorizations, and show that they exist over an appropriate “partially resolved” deformation space for rational double points of types A and D. As a consequence, all simple flops of lengths 1 and 2 can be described in terms of blowups defined from matrix factorizations. We also formulate conjectures which would extend these results to rational double points of type E and simple flops of length greater than 2. The structure of birational maps between algebraic varieties becomes increasingly complicated as the dimension of the varieties increases. There is no birational geometry to speak of in dimension one: if two complete algebraic curves are birationally isomorphic, then they are biregularly isomorphic. In dimension two we encounter the phenomenon of the blowup of a point, and every birational isomorphism can be factored into a sequence of blowups and blowdowns. In dimension three, however, we first encounter birational maps which are biregular outside of a subvariety of codimension two (called the center of the birational map). When the center has a neighborhood with trivial canonical bundle, the birational map is called a flop. The focus of this Received March 26, 2007 and, in revised form, September 3, 2008 and November 2, 2010. The first author was supported by a National Science Foundation Graduate Research Fellowship and by National Science Foundation grant DMS-9983320; the second author was supported by National Science Foundation grants PHY-9907949 and DMS-0301476, the Clay Mathematics Foundation, the John Simon Guggenheim Memorial Foundation, the Kavli Institute for Theoretical Physics (Santa Barbara), and the Mathematical Sciences Research Institute (Berkeley) during the preparation of this paper. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. c ©2013 University Press, Inc. Reverts to public domain 28 years from publication

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تاریخ انتشار 2006